A278618 - OEIS

%I #19 Jun 06 2017 23:30:42

%S 1,0,5,7,45,121,533,1800,7157,26239,101640,384583,1483925,5693247,

%T 22013059,85076183,330014421,1281349195,4985766650,19422653367,

%U 75775163028,295953650376,1157212653030,4529183513913,17743019073381,69565441895001

%N a(n) = Sum_{j=0..n/2} binomial(n-j-1,n-2*j)*binomial(2*n+1,j).

%H G. C. Greubel, <a href="/A278618/b278618.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: x*A(x)/C(x)*B(C(x)), where

%F A(x) = (12-4/sqrt(1-4*x))/(8*sqrt(12*x+2*sqrt(1-4*x)+2))+1/(2*sqrt(1-4*x)),

%F B(x) = 1/((x+1)*sqrt(-3*x^2-2*x+1)),

%F C(x) = sqrt(12*x+2*sqrt(1-4*x)+2)/4-sqrt(1-4*x)/4-1/4.

%F a(n) ~ (1 - 1/sqrt(5)) * 4^n / sqrt(Pi*n). - _Vaclav Kotesovec_, Nov 24 2016

%F a(n) = (2*n + 1)*3F2(1-n/2,3/2-n/2,-2*n; 2,2-n; 4) for n>1. - _Ilya Gutkovskiy_, Nov 24 2016

%F Conjecture: 2*n*(5*n-8)*(2*n-1)*(n+1)*a(n) -n*(115*n^3-344*n^2+299*n-82)*a(n-1) -4*(2*n-1)*(5*n^3+27*n^2-74*n+30)*a(n-2) +36*(n-2)*(5*n-3)*(2*n-1)*(2*n-3)*a(n-3)=0. - _R. J. Mathar_, Dec 02 2016

%t Table[Sum[Binomial[n - j - 1, n - 2*j]*Binomial[2*n + 1, j], {j, 0, n/2}], {n,0,50}] (* _G. C. Greubel_, Jun 06 2017 *)

%o (Maxima)

%o A(x):=(12-4/sqrt(1-4*x))/(8*sqrt(12*x+2*sqrt(1-4*x)+2))+1/(2*sqrt(1-4*x));

%o B(x):=1/((x+1)*sqrt(-3*x^2-2*x+1));

%o C(x):=sqrt(12*x+2*sqrt(1-4*x)+2)/4-sqrt(1-4*x)/4-1/4;

%o taylor(x*A(x)/C(x)*B(C(x)),x,0,20);

%o (PARI) for(n=0,25, print1(sum(j=0,n, binomial(n-j-1,n-2*j)*binomial(2*n+1,j)), ", ")) \\ _G. C. Greubel_, Jun 06 2017

%Y Cf. A005043, A055113.

%K nonn

%O 0,3

%A _Vladimir Kruchinin_, Nov 23 2016